The 29-year-old German, who had been beaten by Hamilton in all four of this year's 'flyaway' season-opening races, led from the 16th pole position of his career all the way to the chequered flag, his supremacy interrupted only by his pit-stops.
It was his first win since last November's Brazilian Grand Prix and the ninth of his career, and trimmed defending champion Hamilton's lead from 27 points to 20.
Hamilton, who slipped from second to third at the start, fought back with daring and great skill, coupled to a three-stop strategy, to finish behind Rosberg and deliver another one-two for the dominant Mercedes team.
He said: "I just clearly got a bad start, got lots of wheel spin, but I'm grateful to get back up to second place. I haven't had such a bad start for a long time.
"There's a long way to go in the championship so, if this is a difficult weekend for me, I'll definitely take it. Today was really a damage limitation exercise for me."
Hamilton added that it was virtually impossible for him to overtake during the race because of the nature of the circuit. "It didn't matter what you did, you could never get near enough and that's a shame for this track."
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